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Theorem elpw1111c 4158

Description: Membership in ℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁1_c. (Contributed by SF, 24-Jan-2015.)

**Assertion**
Ref	Expression
elpw1111c	⊢ (A ∈ ℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁1_c ↔ ∃x A = {{{{{{{{{{{{x}}}}}}}}}}}})

Distinct variable group: x,A

Proof of Theorem elpw1111c Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elpw1 4145	. 2 ⊢ (A ∈ ℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁1_c ↔ ∃y ∈ ℘₁ ℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁1_cA = {y})
2		df-rex 2621	. . . 4 ⊢ (∃y ∈ ℘₁ ℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁1_cA = {y} ↔ ∃y(y ∈ ℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁1_c ∧ A = {y}))
3		elpw1101c 4157	. . . . . . 7 ⊢ (y ∈ ℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁1_c ↔ ∃x y = {{{{{{{{{{{x}}}}}}}}}}})
4	3	anbi1i 676	. . . . . 6 ⊢ ((y ∈ ℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁1_c ∧ A = {y}) ↔ (∃x y = {{{{{{{{{{{x}}}}}}}}}}} ∧ A = {y}))
5		19.41v 1901	. . . . . 6 ⊢ (∃x(y = {{{{{{{{{{{x}}}}}}}}}}} ∧ A = {y}) ↔ (∃x y = {{{{{{{{{{{x}}}}}}}}}}} ∧ A = {y}))
6	4, 5	bitr4i 243	. . . . 5 ⊢ ((y ∈ ℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁1_c ∧ A = {y}) ↔ ∃x(y = {{{{{{{{{{{x}}}}}}}}}}} ∧ A = {y}))
7	6	exbii 1582	. . . 4 ⊢ (∃y(y ∈ ℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁1_c ∧ A = {y}) ↔ ∃y∃x(y = {{{{{{{{{{{x}}}}}}}}}}} ∧ A = {y}))
8	2, 7	bitri 240	. . 3 ⊢ (∃y ∈ ℘₁ ℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁1_cA = {y} ↔ ∃y∃x(y = {{{{{{{{{{{x}}}}}}}}}}} ∧ A = {y}))
9		excom 1741	. . . 4 ⊢ (∃y∃x(y = {{{{{{{{{{{x}}}}}}}}}}} ∧ A = {y}) ↔ ∃x∃y(y = {{{{{{{{{{{x}}}}}}}}}}} ∧ A = {y}))
10		snex 4112	. . . . . 6 ⊢ {{{{{{{{{{{x}}}}}}}}}}} ∈ V
11		sneq 3745	. . . . . . 7 ⊢ (y = {{{{{{{{{{{x}}}}}}}}}}} → {y} = {{{{{{{{{{{{x}}}}}}}}}}}})
12	11	eqeq2d 2364	. . . . . 6 ⊢ (y = {{{{{{{{{{{x}}}}}}}}}}} → (A = {y} ↔ A = {{{{{{{{{{{{x}}}}}}}}}}}}))
13	10, 12	ceqsexv 2895	. . . . 5 ⊢ (∃y(y = {{{{{{{{{{{x}}}}}}}}}}} ∧ A = {y}) ↔ A = {{{{{{{{{{{{x}}}}}}}}}}}})
14	13	exbii 1582	. . . 4 ⊢ (∃x∃y(y = {{{{{{{{{{{x}}}}}}}}}}} ∧ A = {y}) ↔ ∃x A = {{{{{{{{{{{{x}}}}}}}}}}}})
15	9, 14	bitri 240	. . 3 ⊢ (∃y∃x(y = {{{{{{{{{{{x}}}}}}}}}}} ∧ A = {y}) ↔ ∃x A = {{{{{{{{{{{{x}}}}}}}}}}}})
16	8, 15	bitri 240	. 2 ⊢ (∃y ∈ ℘₁ ℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁1_cA = {y} ↔ ∃x A = {{{{{{{{{{{{x}}}}}}}}}}}})
17	1, 16	bitri 240	1 ⊢ (A ∈ ℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁1_c ↔ ∃x A = {{{{{{{{{{{{x}}}}}}}}}}}})

Colors of variables: wff setvar class

Syntax hints: ↔ wb 176 ∧ wa 358 ∃wex 1541 = wceq 1642 ∈ wcel 1710 ∃wrex 2616 {csn 3738 1_cc1c 4135 ℘₁cpw1 4136

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4079 ax-sn 4088

This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2479 df-ne 2519 df-rex 2621 df-v 2862 df-nin 3212 df-compl 3213 df-in 3214 df-un 3215 df-dif 3216 df-ss 3260 df-nul 3552 df-pw 3725 df-sn 3742 df-1c 4137 df-pw1 4138

This theorem is referenced by: ltfinex 4465

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